Christmas problem 2004

This is my third Christmas maths problem in three years. You can win a nice prize with it.

Ten pirates arrive at a space station with a space ship. The station has the shape of a regular tetrahedron. They decide that some of them will settle there, but as far away from each other as possible.

Given a regular tetrahedron with sides of length 1 kilometer, find the positions of n points on the tetrahedron such that the smallest distance between two points out of these n is as great as possible. Do this for n=2,3,4,5,6,7,8,9,10 (nine cases). Distances must be measured along shortest paths over the tetrahedron, not directly through space.
Give exact descriptions and sketches of your configurations, and a concise explanation of the way you found the configurations. Mention in each case the minimal distance between two points in your configuration.

The best solutions of competitors will be published on this website, with the names of the competitors. There is a prize in each of two categories: professionals (default category) and amateurs (only for non-mathematicians).

Send me your answers with your name and address. If you want to be an amateur, just let me know. You get an affirmation of receipt as soon as possible.
Applications close on the 6-th of Januari 2005. On that day, I must have your contribution. After that, I need another two weeks to publish the results.

dr HFH Reuvers, Brusselsestraat 92, 6211 PH Maastricht Nederland

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